The Fundamental Theorem of Algebra

If $a$ is a root of $P(x)=0$, then $(x-a)$ is a factor of $P(x)$, so $\deg(P)\ge1$. If $a,b$ are both roots, $(x-a)(x-b)$ is a factor, so $\deg(P)\ge2$; and in general, if $P(x)=0$ has $n$ (distinct) roots, then $\deg(P)\ge n$. Equivalently: a degree-$n$ polynomial equation cannot have more than $n$ (distinct) roots.

Multiplicity. If $(x-a)^k$ is a factor of $P(x)$ but $(x-a)^{k+1}$ is not, then $a$ is a root of multiplicity $k$. E.g. $3$ is a root of multiplicity $2$ for $x^2-6x+9=0$ and for $x^3-7x^2+15x-9=0$; a root of multiplicity $1$ is called a simple root. Even counting with multiplicity, a degree-$n$ equation still cannot have more than $n$ roots.

The proof is beyond this course's scope, but the theorem is used to prove something stronger and more useful: a degree-$n$ equation has at least $n$ roots in $\mathbb C$ counted with multiplicity — combined with the "at most $n$" fact above, this gives the working statement used throughout the chapter:

A polynomial equation of degree $n$ has exactly $n$ roots in $\mathbb C$, when the roots are counted with their multiplicities. …